When the concept of tensor product is studied from a categorical perspective, it is usually treated either as additional structure on a category - which leads to the theory of monoidal and enriched categories - or in an ad-hoc way involving free algebras of some kind. Quite surprisingly, as yet no internal categorical construction in terms of limits and colimits has been proposed.
The aim of my talk is to do precisely this. I will first state a general construction of such an intrinsic tensor product, based on the work [1, 2, 3] in the context of semi-abelian categories. Then I will give an overview of the main examples and sketch some applications.
This is joint work with Manfred Hartl.
 A. Carboni and G. Janelidze, Smash product of pointed objects in lextensive categories, J. Pure Appl. Algebra 183 (2003), 27-43.
 M. Hartl and B. Loiseau, On actions and strict actions in homological categories, Theory Appl. Categ. 27 (2013), no. 15, 347-392.
 M. Hartl and T. Van der Linden,The ternary commutator obstruction for internal crossed modules, Adv. Math. 232 (2013), no. 1, 571-607.
Tim Van der Linden (Université Catholique de Louvain, Belgium)